16. bis 22. September 2001 in Wien

The probability of the event
of the
multinomial distribution is given by the formula

(1)

where

and

The task of this paper is to find all maxima of (1), usually
called the modes of the multinomial distribution.

Two methods for this exist. One is based on Moran's
inequality, which is not presented in its
sharper version in textbooks like Feller [1957] or Johnson-
Kotz-Balakrishnan [1997]. It is slow for large
values of the number of the . By far more
effective is the idea of Finucan [1964] to consider the
step-function

(2)

The jumps of can be determined quite easily. If
holds,
are the
components of the unique mode. If there are more than one
mode their components can be determined
as
or
.
The subscripts where has to be subtracted
are easy to determine. Numerical examples are given
which exhibit the method.

By a similar method the modes of a hypergeometric
distribution can be calculated.